Geodesic
Existence of discontinuous absolute minima for certain multiple integrals without growth properties
Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 4, pp. 661-671 Cet article a éte moissonné depuis la source Biblioteca Digitale Italiana di Matematica

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In the present paper the author discusses certain multiple integrals $I(u)$ of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals $\mathfrak{I}(u)$, to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals $I(u)$ and $\mathfrak{I}(u)$ are reduced to simpler form $H(v)$ and $\mathcal{H}(v)$ to which the existence theorems above apply. Thus, we derive that $I(u) \le \mathfrak{I}(u)$, $H(v) \le \mathcal{H}(v)$, we obtain the existence of the absolute minimum for the Serrin forms $\mathfrak{I}(u)$ and $\mathcal{H}(v)$, and such minimum is given by BV functions, possibly discontinuous and not of Sobolev. 
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     title = {Existence of discontinuous absolute minima for certain multiple integrals without growth properties},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali},
     pages = {661--671},
     year = {1988},
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Cesari, Lamberto. Existence of discontinuous absolute minima for certain multiple integrals without growth properties. Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 4, pp. 661-671. http://geodesic.mathdoc.fr/item/RLINA_1988_8_82_4_a5/

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